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## Homework Statement

Let [itex]f_k\rightarrow f[/itex] in [itex]L^2(\Omega)[/itex] where [itex]|\Omega|[/itex] is finite. If [itex]\int_{\Omega}{f_k(x)}dx=0[/itex] for all [itex]k=1,2,3,\ldots[/itex], then [itex]\int_{\Omega}{f(x)}dx=0[/itex].

## Homework Equations

## The Attempt at a Solution

I started by playing around with Holder's inequality and constructing examples where this is the case. Since every other example I create usually does not converge to a function. I used functions defined on a bounded symmetric interval like [itex]\dfrac{1}{k}x[/itex], [itex]\sin(\dfrac{1}{k}x)[/itex]. However, I do not see how to actually set up to make this conclusion.